Simple mathematical problem (?)

[LuxFerum]

Did you try googling for 0.999=1 ? There are lots of websites about this. Also, the normal way of showing Cantor's famous diagonal proof has a section explaining that many numbers can be written in two distinct ways, and that therefore any number written with an infinite string of 9s at the end must be replaced by the alternative form ending in a string of 0s.

Those proofs are not good enough.
I need hardcore proofs

 

[ceptimus]

Lux, your counterexample doesn't have a decimal point (or maybe you use a comma) so your number is infinite. but 9.999... isn't, it is exactly equal to 10.

I agree that you can't bring infinities into these sorts of equations. If you do, you can prove anything. But an infinite number of decimal places doesn't make an infinite number.

 

[geni]

Those proofs are not good enough.
I need hardcore proofs

Can you think of any situation where 0.999... and 1 behave differently?

 

[geni]

Lux, your counterexample doesn't have a decimal point (or maybe you use a comma)

LuxFerum is on the european mainland they use commers there.

 

[LuxFerum]

Can you think of any situation where 0.999... and 1 behave differently?

yes
see my all my posts today and you will find it.

 

[LuxFerum]

Lux, your counterexample doesn't have a decimal point (or maybe you use a comma) so your number is infinite. but 9.999... isn't, it is exactly equal to 10.

I agree that you can't bring infinities into these sorts of equations. If you do, you can prove anything. But an infinite number of decimal places doesn't make an infinite number.

Did anyone read my post about thing are not in infinity but things tends to somthing in infinity?
I spend a lot of time in that one, please read.

 

[69dodge]

Look I will try it again.

            999999999999999999999999...  This is infinity
            999999999999999999999999...  This is infinity too


When we multipliy something by ten, the numbers move one row to the left.
And when we multiply infinity by ten, the result is infinity.


           9999999999999999999999999...  This is infinity multiplyed by 10, and still infinite
            999999999999999999999999...  This is infinity 

When we subtry infinite from infinte the result is zero
sooo 


           9999999999999999999999999...  
           -999999999999999999999999...  
           ----------------------------
           9000000000000000000000000...

So by this result is clear that infinity is equal to zero.

If you can find the error in this, you can see the error in you proof

Some operations on series---for example, rearranging the order of terms---are allowed if the series is absolutely convergent but are not allowed otherwise.

The series<blockquote>0.9 + 0.09 + 0.009 + 0.0009 + . . .</blockquote>converges absolutely but the series<blockquote>9 + 90 + 900 + 9000 + . . .</blockquote>diverges.

Subtracting 0.9999. . . from 9.999. . . to get 9 is essentially rearranging terms:<blockquote>(9 + 0.9 + 0.09 + . . .) - (0.9 + 0.09 + 0.009 + . . .) =
9 + (0.9 - 0.9) + (0.09 - 0.09) + (0.009 - 0.009) + . . .</blockquote>

 

[Walter Wayne]

Yes there is something.
That something tends to zero, But I can multiply it by something that tends to infinity faster than that something is going to zero.

Showing that something tends to infinity faster than something else tends to 0, does not show anything.

1 * Infinity = Infinity
0.99999 * Infinity = Infinity
0 * Infinity is undefined and can, depending on the situation take on any value on the range [0, infinity )

I read your long proof, and it only prooves that exponential series tend to infinity, faster then geometric. Congrats on using L'Hopital's Rule.

Walt

 

[Walter Wayne]

9 or 9.000...something?
I can make that something grow up faster to every way you can made it smaller.

You can't have 0...something! In the series of digitis .000..., every zero has another zero following it. So you can't find a final zero append a "something" to.

 

[LuxFerum]

You can't have 0...something! In the series of digitis .000..., every zero has another zero following it. So you can't find a final zero append a "something" to.

you are doing the same thing again.

You use the infinity to get another number and them you take the infinity off.

That is wrong.

 

[davefoc]

I read through most of this long thread. It seems that this is both a philosophical question and a mathematical question which is why I lked Earthborn's (and others) argument.

Since a number doesn't exist between 1 and .9... they are the same.

Perhaps not as rigorous as some of the other proofs but complete rigor isn't needed for an engineer to be happy.

 

[LuxFerum]

Showing that something tends to infinity faster than something else tends to 0, does not show anything.

1 * Infinity = Infinity
0.99999 * Infinity = Infinity
0 * Infinity is undefined and can, depending on the situation take on any value on the range [0, infinity )

I read your long proof, and it only prooves that exponential series tend to infinity, faster then geometric. Congrats on using L'Hopital's Rule.

Walt

It proves that sum 9*10^(-n) is not 1 in infinity.

That is what Im trying to prove there.

That sum just tend to 1.

If that sum is 1 in infinity then

sum 9*10^(-n) - sum 2^(-n) "is" zero.
but that is not true.
it tends to zero.

 

[69dodge]

Originally posted by LuxFerum
Did anyone read my post about thing are not in infinity but things tends to somthing in infinity?
I spend a lot of time in that one, please read.

I read it. You clearly understand how to work with infinite series. So any disagreement is simply a disagreement about definitions. The definition I have always seen for the meaning of decimal expansions says that "0.999. . ." means "the limit of the sequence 0.9, 0.99, 0.999, . . ." In other words, you consider the sequence of partial sums, you find its limit---which is a single number, and then you forget about how you got that number.

The sequence 0.9, 0.99, 0.999, . . . approaches 1. In other words, the limit of that sequence equals 1. If "0.999. . ." is defined to denote that limit, then it too equals 1.

 

[TillEulenspiegel]

Sorry I'm late to the party and I havent read the whole thread,but how's this:
A suggested ‘proof’ is
X=0.999(recurring)
10X=9.9999
10X-X = 9X <-------------- Heres the prob
9X=9X
X=1

Hence 0.999(recurring)=1


ummmm
X=0.999(recurring)
10X=9.9999( recurring)
10X-X = 9. <---- as we already know the value of x
9X=9X= 9* .99999(recurring)
X=.999 (recurring)
You arbitrarily assigned x the value of 1 by designating it a a whole number variable after it was assigned the value of .999...towards infinity The fact that were dealing in factors of 10 and 1 in the proof is actually easier then the next digression towrds x
10x=9.99(recurring)
10 x - x = 9
ummmmmm
9x - x = 8*x Ya thats it! HEHEHE

There is also a therem that states that there are more numbers between 1 and 2 then between 0 and 1 now theres some infinities for you entertainment.
P.S. not a math wiz

 

[LuxFerum]

I read it. You clearly understand how to work with infinite series. So any disagreement is simply a disagreement about definitions. The definition I have always seen for the meaning of decimal expansions says that "0.999. . ." means "the limit of the sequence 0.9, 0.99, 0.999, . . ." In other words, you consider the sequence of partial sums, you find its limit---which is a single number, and then you forget about how you got that number

So you agree that 0,999.. tends to 1 and is not equal to 1?

 

[Walter Wayne]

You cant work with infinity that way.

Like Walter Wayne post

Originally posted by Walter Wayne

And a fun one for playing with infinities:
0=(1-1)+(1-1)+(1-1)+ ...
=1 + (-1+1) + (-1+1) + ...
=1

You cant use infinity to make a nine apears and then take the inifity off.

Actually, that quote you give has to do with arguments about the eveness or oddness of infinity, and has nothing to do with that.

10*0.999... = 9.999... = 9 + 0.999..., you can subtract 0.999... from that.

 

[LuxFerum]

Sorry I'm late to the party and I havent read the whole thread,but how's this:
A suggested ‘proof’ is
X=0.999(recurring)
10X=9.9999
10X-X = 9X <-------------- Heres the prob
9X=9X
X=1

Hence 0.999(recurring)=1


ummmm
X=0.999(recurring)
10X=9.9999( recurring)
10X-X = 9. <---- as we already know the value of x
9X=9X= 9* .99999(recurring)
X=.999 (recurring)
You arbitrarily assigned x the value of 1 by designating it a a whole number variable after it was assigned the value of .999...towards infinity The fact that were dealing in factors of 10 and 1 in the proof is actually easier then the next digression towrds x
10x=9.99(recurring)
10 x - x = 9
ummmmmm
9x - x = 8*x Ya thats it! HEHEHE

There is also a therem that states that there are more numbers between 1 and 2 then between 0 and 1 now theres some infinities for you entertainment.
P.S. not a math wiz

this have the same mistake as this one

            999999999999999999999999...  This is infinity
            999999999999999999999999...  This is infinity too


When we multipliy something by ten, the numbers move one row to the left.
And when we multiply infinity by ten, the result is infinity.


           9999999999999999999999999...  This is infinity multiplyed by 10, and still infinite
            999999999999999999999999...  This is infinity 

When we subtry infinite from infinte the result is zero
sooo 


           9999999999999999999999999...  
           -999999999999999999999999...  
           ----------------------------
           9000000000000000000000000...

So by this result is clear that infinity is equal to zero.

 

[LuxFerum]

You cant use infinity to make a nine apears and then take the inifity off.

Actually, that quote you give has to do with arguments about the eveness or oddness of infinity, and has nothing to do with that.

10*0.999... = 9.999... = 9 + 0.999..., you can subtract 0.999... from that. [/B][/QUOTE]
no you cant.

look at this
infinity - infinity =0

And whe know that
9+infinity = infinity

Then whe got

9+infinity-infinity=0

9=0

You just cant work with infinity like that.

 

[ceptimus]

this have the same mistake as this one

            999999999999999999999999...  This is infinity
            999999999999999999999999...  This is infinity too

The mistake is right there. You can't have a leftmost digit when you write infinity, so you would have to put something like:

...9999999999999999...

Note: there are three dots at both ends.

And then the rest of your proof falls apart.

 

[xouper]

LuxFerum: Those proofs are not good enough.
I need hardcore proofs

Several people, including myself, already posted a hardcore proof using the geometric series theorem. That is a rigorous and airtight mathematical proof. If you do not accept that, then you are not doing mathematics, but are instead playing in your own sandbox by yourself.

Also, regarding your comments about 999... with an infintie number of nines going to the left of the decimal point, that notation is not a real number, so your argument doesn't apply to a real number such as 0.999... with the nines extending to the right of the decimal point (or comma for those who like that better).

Also, your "analysis" of infinity and "tends to" completely misses the point of the definition of the notation 0.999... and so I will ask again, please stop trying to redefine the standard notation using your own mistaken notions.

The notation 0.999... means that all the nines are already there all at one time. It is not a process of tacking another nine on the end until you reach infinity. It is not a process that "tends" toward 1. It is just another way of representing the number 1.

Also, if you continue to insist that 0.999... does not equal 1, then please prove your assertion by answering the questions posed to you previously, which I will repeat here (again):

• Find a real number N such that N is less than 1 and N is greater than 0.999...
  
  or
  
• Find two positive integers A and B such that A/B = 0.999... and A does not equal B.

When you claim that 0.999... does not equal 1, then you are also claiming both of the above assertions, since they are mathematically equivalent to the claim that 0.999... is not equal to 1. Please address these questions and stop wasting our time with all that other handwaving you've been doing.